Kinetic energy fix for low internal energy flows

When the kinetic energy of a flow is dominant, numerical schemes employed can encounter difficulties due to negative internal energy. A case study with several commonly used conservative schemes (MUSCL, ENO, WENO and CE/SE) shows that high order schemes may have less ability to preserve positive internal energy (MUSCL and CE/SE), or present less accurate results (WENO and ENO) when the internal energy to kinetic energy ratio is low. By analyzing the positivity property for second-order conservative schemes with large fixed CFL number conditions for time step restriction, this paper proposes the energy consistency conditions for second-order Riemann-solver type schemes and CE/SE method. According to the said energy consistency conditions, a kinetic energy fix method which limits the magnitude of kinetic energy relative to the total energy is introduced. The numerical examples show that the kinetic energy fixed CE/SE method produces reasonable results and keeps positive internal energy for flows with very low internal energy even when a vacuum occurs.